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Propagation of spatial imprecision in imprecise
quantitative data in agronomy
Karima Zayrit1, Eric Desjardin
1, Cyril de Runz
1 & Herman Akdag
2
1 CReSTIC, Université de Reims Champagne-Ardenne
[email protected], [email protected], [email protected] 2 LIP6, Université Paris 6
Abstract
One of the stakes of Observox, an observatory of agricultural practices, is to deal
with imperfect spatial information and to always associate a quality evaluation to
acquired or computed data. So, we introduce the notion of fuzzy geographical
entities. Then, we consider both spatial and quantitative information in order to
obtain fuzzy local quantitative information. This paper proposes a new operator
which gives the fuzzy quantity of spatially disseminated chemical products for each
location.
Keywords: Imprecision, fuzziness, propagation, agriculture.
1 Introduction
In the past 30 years, the use of GIS has grown and today it is the standard for
managing spatial – as located on Earth – and spatiotemporal data. Their use goes
from archaeology (Conolly and Lake, 2006; De Runz and Desjardin, 2010) to
agronomy (the context of this work).
The spatial feature of studied entities is often as imprecise (and/or uncertain) as
its quantitative and descriptive features. According to literature (Klir and Yuan,
1995; Smets, 1995; Fisher et al., 2006; De Runz et al., 2008), fuzzy set theory and
fuzzy logic are a good approach to deal with this kind of data imperfection. Then,
one can build entities where both the spatial and the quantitative features are fuzzy.
The fuzzy set theory allows overlap between fuzzy shapes. The question is: what
is the value of fuzzy quantitative attributes in a location where two or more fuzzy
spatial shapes overlap? The answer to this question is the heart of this article.
Actually, in order to build an observatory on agricultural practice in the Vesle
Basin, we have to deal with multiple sources of information that introduce impreci-
sion in the object. From this situation, spatial and quantitative information may thus
be imprecise.
Indeed, in the spatial context, there is two main ways for modelling imprecision
(Bejaoui et al., 2009). In the first hand, the crisp models extend or transform pre-
cise spatial concept in order to represent spatial imprecision as for instance the
Egg-Yolk model (Cohn and Gotts, 1996). In the second hand, the models are based
on uncertain mathematical theories as the ones, such as (Navratil, 2007), using
fuzzy sets (Zadeh, 1965), either those, for example (Worboys, 1998), exploiting
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rough sets, or those, as for instance (Pfoser et al., 2005), using probabilities.. The
fuzzy models give us a unique and soft framework that allows us to represent im-
precision and to better conceptualize the reality (see Smets (1995)). As the aim of
our system is to give interpretable information in each location of the monitored
space, fuzzy data modeling data is used by us.
This paper exposes our opinion and choices in order to answer to these questions
in the context of agronomical data exploitation. It introduces an operator for the
propagation of spatial imprecision to imprecise quantitative information. It also
presents a global structure for the management of fuzzy geo-entities. This structure
is based on a fuzzy data storage impact analysis.
Section 2 is devoted to the imprecise geo-entity modelling in the framework of
fuzzy set theory. Then, the propagation of imprecision in overlapping areas is stud-
ied (section 3). Finally, the conclusion is presented in section 4.
2 Fuzzy modeling of agronomical entities
In the sustainable development context, the AQUAL project (a State-Region
Project in the Champagne-Ardenne, France) highlights the need of a monitoring
environment for the study of agricultural practices and their pressure on the water
resources in the Vesle basin. It is called Observox and it exploits data coming from
heterogeneous sources: satellite images, land registry, statistical data, Corine Land
Cover and other European data. The construction of a unique set of entities implies
the combination of information coming from all the sources. The built entities thus
induce some imprecision in the definition of spatial features and quantitative attrib-
utes (Shi, 2010).
On the other hand, Fisher in (Ficher, 1996) presents a comparative study be-
tween crisp sets and fuzzy sets in order to model landscape. The formers simplify
the modeling but could amplify errors. The latters make the models and the treat-
ments more complex. In (Fisher et al, 2006), the authors present a taxonomy of
uncertainty in spatial context where the vagueness is associated to the fuzzy set
theory. According to (Duckham et al., 2001), vagueness is a special type of impre-
cision. Vagueness and imprecision could be both represented by fuzzy sets (Bou-
chon-Meunier, 1995; Klir and Yuan, 1995; Smets, 1995) introduced in (Zadeh,
1965).
According to this, in agronomical studies as well as in geography, the geograph-
ical entities could be modeled as fuzzy geographical entities. Those entities have a
label, a fuzzy spatial shape and a set of fuzzy quantities (each quantity corresponds
to a specific attribute such as population or a specific chemical). The definition of a
geographical entity may be defined as follows.
Let Ω be the set of studied geographical entities A1,…,An. Let be Ϙ the set of
monitoring quantitative information (Q1,…,Qm) if one supervises m different infor-
mation (P1,…,Pm) as for instance m different molecules or products . Let us define a
fuzzy geographical entity Ai in Ω as an object described by:
- A label or concept LAi member of an ontology.
- A fuzzy set FSAi describing its spatial representation. The membership
function µSAi of FSAi is defined on ℝ2.
- A fuzzy quantity FQjAi for each quantity Qj (of Pj) in Ϙ. The member-
ship function µQjAi of FQjAi is defined on ℝ+.
An example of an Ai is shown in Table 1.
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Table 1. A fuzzy geographical entity Ai (only one quantitative information is shown)
Type of information Attribute Value example
Label/Function LAi Vineyard
Spatial definition FSAi
Quantity of Pj
(e.g. Isoproturon)
FQjAi
If Qj is a precise quantity (with a value a), it could be represented by a singleton
in the fuzzy set theory as follows: if q=a then µQjAi(q)=1 else µQjAi(q)=0 such as q
belongs to ℝ+). This principle is presented in figure 1.
Figure 1. Illustration of a precise quantity a represented in the fuzzy set theory. In the context of OBSERVOX, Ϙ is the set of studied chemical (or at a micro-
scale, the set of phytosanitary molecules). It could be for example a fuzzy pre-
scribed dose or an estimation of quantity which was actually spread.
The next section is devoted to the sensibility of quantity values in a space loca-
tion.
3 Propagation of imprecision
Let us consider x a location. We consider that the confidence in FQjAi should be
put into perspective with the membership degree ( ) in order to define
FQjAi,x with its membership function µQjAi,x as proposed in (1). In this definition,
when a fuzzy geographical entity does not participate to the definition of x
( ( ) ), the quantity of product Pj diffused at x by is certain and null. ( ) Q A , ( ) ( ( ) A ( ))
Q A , ( ) Q A , ( )
(1)
with q in ℝ+ and T an aggregation function, usually a t-norm such as the multi-
plication or the minimum.
The imprecision, conceptualized using a classical fuzzy number for quantities
and by fuzzy area for spatial feature, is the propagated in the consideration of fuzzy
quantities at a specific location. As our goal is to consider all the quantities of a
specific product at each location of the space, an aggregation operator is now need-
ed for obtaining the combined information. Then, we use the Zadeh„s extension
0 q
1
a
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principle that allows to extend usual operation in the fuzzy set context such as in
our context the sum (due to the additive aspect of product diffusion).
Thus if we deal with an additive information Pj, using this hypothesis and Za-
deh‟s extension principle we define FQj,x the overall quantity at the position x by
following the equation (2) for the definition of its membership function Q , .
( ) ( ( Q A , ( ) Q A , ( ))) (2)
Figure 2. Illustration of the imprecision propagation on quantitative value Qj of Pj for a
specific location x, with Ω=A1,A2, (x)=0.8 and (x)=0.4.
In order to test the feasibility of our approach, we illustrate it using two over-
lapped fuzzy geographic entities (A1 and A2) at a specific location x (figures 2 and
3). The goal is in this example to determine the total quantity of a chemical Pj (cor-
responding to Bentazone) at x.
This principle allows us to compute the quantity of each monitored molecule in
every location of the studied region. The confidence in the computed fuzzy quantity
is lower (or equal) than the original confidence in each fuzzy geographical entities.
Figure 3. Illustration of the imprecision propagation: a spatial/quantity view.
At x, 𝜇 (x)=0.8 and 𝜇 (x)=0.4.
𝜇Q A1 , 𝜇Q A2 , 𝜇Q ,
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4 Conclusion
In this paper, we propose a study of the imprecision propagation from spatial in-
formation to quantitative one. We firstly introduced our context and our approach
of a fuzzy geographical entity. Next, we proposed a new operator of imprecision
propagation.
This paper is a starter for the future construction of an agricultural practice ob-
servatory. In our future work, we will use conceptual approach that allows us to
automatically obtain a fuzzy spatiotemporal data storage solution (Zoghlami et al.,
2011), but we also want to study the propagation of quantitative imprecise infor-
mation into other topological relations between fuzzy spatial objects.
This paper is a preliminary study before building the observatory. It presents our
choice at the beginning of the project. In our future work, we will develop our ap-
proach by defining new fuzzy agronomical indices in the observatory.
Acknowledgements
We would thank the Seine-Normandy Water Agency, Champagne-Ardenne Re-
gion Council, France and European Union, through the FEDER, for their funding
of the project CPER AQUAL.
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